g^{'} = g \frac{ρ _{1} - ρ _{2}}{ρ _{2}},

and use the Boussinesq limit

ρ_{1} ≃ ρ_{2}

except in buoyancy. The balances become

g_{t}^{'} + u g_{x}^{'} = - g^{'} \frac{w _{e}}{h},

h_{t} + u h_{x} + h u_{x} = w_{e} - w_{d},

u_{t} + u u_{x} + g^{'} h_{x} + \frac{h}{2} g_{x}^{'} = - u \frac{w _{e}}{h} .

Thus, for

q = (g^{'}, h, u)^{T}

q_{t} + u 0 h /2 0 u g^{'} 0 h u q_{x} = - g^{'} w_{e} / h w_{e} - w_{d} - u w_{e} / h,

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